Dynamical Mechanisms and Energy Conversion in Rayleigh-Bénard Convection

WANG He-yuan, HE Xiang-hong, YANG Guang

doi:10.13371/j.cnki.chin.q.j.m.2026.02.001

Chinese Quarterly Journal of Mathematics >

2026 , Vol. 41 >Issue 2: 111 - 127

DOI: https://doi.org/10.13371/j.cnki.chin.q.j.m.2026.02.001

Dynamical Mechanisms and Energy Conversion in Rayleigh-Bénard Convection

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1. Guangdong University of Science and Technology, Dongguan 523083, China; 2. Joint Innovation Industry College, Quanzhou Vocational and Technical University, Jinjiang 362268, China

WANG He-yuan (1963-), male, native of Shenyang, Liaoning, professor of Guangdong University of Science and Technology, Ph.D., engages in nonlinear dynamics; HE Xiang-hong (1978-), female, native of Shandong, engineer of Guangdong University of Science and Technology, Ph.D., engages in mechanical analysis; YANG Guang (1964-), female, native of Shenyang, Liaoning, professor of Quanzhou Vocational and Technical University, Ph.D., engages in intelligent control theory and its applications.

Received date: 2025-08-30

Online published: 2026-06-30

Supported by

Supported by National Natural Science Foundation of China (Grant No. 11572146); Research Projects Foundation of Guangdong University of Science and Technology (Grant No. GKY-2025KYZDK-25); The PhD Research Startup Foundation of Guangdong University of Science and Technology (Grant No. GKY-2022BSQD-36); Guangdong University of Science and Technology Quality Engineering Project (Grant No. GKZLGC2025268).

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Abstract

Numerous studies have addressed the stability of fluid between two thermally conducting plates (frequently abbreviated as Rayleigh-B´enard convection). The Lorenz system serves as the classical model of Rayleigh-B´enard convection problems and provides a paradigm for the laminar-to-turbulent transition. In this paper we study the dynamical mechanism and energy conversion of the Lorenz equation. The Lorenz chaotic system is transformed into a Kolmogorov-type system, which is decomposed into four types of torques: inertial torque, internal torque, dissipation torque and external torque. By combining different torques, the key factors for the generation of chaos in the Lorenz
system – the mathematical model corresponding to the Rayleigh-B´enard convection problem – have been studied. We further investigate the conversion among Hamiltonian, kinetic and potential energies, as well as the correlation between the energies and the Reynolds number. It is concluded that the combination of all four torques is necessary
to produce chaos, and the system can produce chaos only when the dissipative torques match the driving (external) torques. Any combination of three types of torques cannot produce chaos. The external torque, driven by heat from the bottom plate, supplies energy and that leads to the production of roll vortices and chaos. Moreover, we introduce
the Casimir function to analyze the system dynamics, and use its derivative to formulate the energy conversion. The bound of the chaotic attractor is obtained by the Casimir function and Lagrange multiplier. It is found that the Casimir function reflects the energy conversion and the distance between the orbit and the equilibria.

Key words： Rayleigh-Bénard convection; Dynamical Mechanism; Kolmogorov system; Chaos

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WANG He-yuan, HE Xiang-hong, YANG Guang . Dynamical Mechanisms and Energy Conversion in Rayleigh-Bénard Convection[J]. Chinese Quarterly Journal of Mathematics, 2026 , 41(2) : 111 -127 . DOI: 10.13371/j.cnki.chin.q.j.m.2026.02.001

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